Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-14: (a) Find the distance \(d(A, B)\) between \(A\) and \(B\). (b) Find the midpoint of the segment \(A B\). $$ A(-4,7), \quad B(0,-8) $$

Step-by-Step Solution

Answer

The distance is \(\sqrt{241}\) and the midpoint is \((-2, -\frac{1}{2})\).

Step 1: Find the difference in coordinates

To find the distance between points \(A\) and \(B\), we first need the differences in the \(x\) and \(y\) coordinates. Given the coordinates \(A(-4, 7)\) and \(B(0, -8)\), calculate the difference in \(x\)-coordinates: \( \Delta x = 0 - (-4) = 4\). Calculate the difference in \(y\)-coordinates: \( \Delta y = -8 - 7 = -15\).

Step 2: Use the distance formula

The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substitute the differences into this formula: \( d = \sqrt{4^2 + (-15)^2} = \sqrt{16 + 225} = \sqrt{241} \).

Step 3: Calculate the midpoint coordinates

To find the midpoint of the segment \(AB\), use the midpoint formula: \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute the given coordinates: \( M = \left( \frac{-4 + 0}{2}, \frac{7 + (-8)}{2} \right) = \left( -2, -\frac{1}{2} \right) \).

Key Concepts

midpoint formula

To find the midpoint of a line segment connecting two points, we use the midpoint formula. This formula helps us calculate the coordinates of a point that divides the segment into two equal halves. The midpoint ensures that the line segment is perfectly balanced.The formula for determining the midpoint, \( M(x, y) \), given two endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \), is given by:

• \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

By adding the x-coordinates and y-coordinates separately and dividing each by two, we can find the exact center between the two points. For example, if we have points \( A(-4,7) \) and \( B(0,-8) \), plug these values into the formula:

• \( M = \left( \frac{-4 + 0}{2}, \frac{7 + (-8)}{2} \right) = \left( -2, -\frac{1}{2} \right) \)

This midpoint \( (-2, -\frac{1}{2}) \) is the precise point that lies halfway between \( A \) and \( B \). Understanding how to calculate a midpoint is essential for various applications in geometry and real-life scenarios.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses numbers to represent geometric shapes and their properties. It allows us to analyze and solve geometric problems using a coordinate plane where each point is represented by a pair of numbers (coordinates), typically as \((x, y)\).This approach enables clear and precise calculations of distances, angles, and other geometric properties. With coordinate geometry, one can easily:

• Determine distances between points using the distance formula.
• Find midpoints of line segments using the midpoint formula.
• Identify relationships and intersections of lines, curves, and figures.

In the context of coordinate geometry, each point on the plane corresponds to specific coordinates, allowing for a detailed study and application of algebraic techniques to geometry. This foundation supports a deeper understanding and efficient problem-solving method for various mathematical concepts.

distance between points

To calculate the distance between two points on a coordinate plane, we utilize the distance formula. This formula derives from the Pythagorean theorem and is essential for finding the exact length of the segment that connects the two points.Assuming points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between these points is given by:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This equation calculates the change in x (\(\Delta x\)) and the change in y (\(\Delta y\)), squares these values, sums them up, and finally takes the square root to provide the distance. For example, with points \(A(-4,7)\) and \(B(0,-8)\), compute it as follows:

• \(\Delta x = 0 - (-4) = 4\)
• \(\Delta y = -8 - 7 = -15\)
• \(d = \sqrt{4^2 + (-15)^2} = \sqrt{16 + 225} = \sqrt{241}\)

The calculated distance, \(\sqrt{241}\), represents the exact length between points \(A\) and \(B\). This method is widely applicable in various fields, from mathematics to physics and beyond, whenever precise distance calculations are necessary.